On strongly multiplicative sets

A multiplicative subset $S$ of a ring $R$ is called \textit{strongly multiplicative} if $(\bigcap_{i\in\Delta}s_iR)\cap S \neq \emptyset$ for each family $(s_i)_{i\in\Delta}$ of elements in $S$. In this paper, we investigate how these sets help stabilize localization and ideal operations. We show that localization and arbitrary intersections commute, meaning $S^{-1}(\bigcap I_\alpha) = \bigcap S^{-1}I_\alpha$ for any family of ideals, if and only if $S$ is strongly multiplicative. Furthermore, we characterize some important classes of rings, such as total quotient rings and strongly zero-dimensional rings, in terms of strongly multiplicative sets. We also answer an open question by Hamed and Malek about whether this condition is needed for the existence of $S$-minimal primes. Furthermore, we demonstrate that if $S$ is a strongly multiplicative set and $S \not\subseteq U(R)$, then $S$-minimal primes are not classical prime ideals, and we provide an algorithmic approach to constructing such ideals. Finally, we prove a Strong Krull's Separation Lemma, which guarantees a maximal ideal disjoint from $S$. As an application of Strong Krull's Separation Lemma, we establish a one-to-one correspondence between the maximal ideals of $S^{-1}R$ and the maximal ideals of $R$ disjoint from a strongly multiplicative set $S$ of $R$.